Recently, a lot of attention has been focused on the study of the following Choquard-type Hamiltonian system

{ − Δ u + u = ( I μ 1 ∗ | v | p | x | α ) | v | p − 2 v | x | α ,     in     ℝ N \ { 0 } , − Δ v + v = ( I μ 2 ∗ | u | q | x | β ) | u | q − 2 u | x | β ,     in     ℝ N \ { 0 } , u ( x ) , v ( x ) → 0,   when       | x | → ∞ , (1.1)

where N ≥ 3 , 0 < μ 1 , μ 2 < N , 0 ≤ α ≤ μ 1 2 , 0 ≤ β ≤ μ 2 2 , p , q > 1 , I μ i = 1 | x | N − μ i for i = 1 , 2 , ∗ is the convolution in ℝ N .

When α = β = 0 , μ 1 = μ 2 , p = q , u = v , (1.1) reduces to the following classic Choquard equation

− Δ u + u = ( I μ 1 ∗ | u | p ) | u | p − 2 u ,     in     ℝ N . (1.2)

Equation (1.2) has a physical prototype, as pointed out by Lieb [1] , Choquard introduced the equation

i ψ t = − Δ ψ − ( I 2 ∗ | ψ | 2 ) ψ ,   ( x , t ) ∈ ℝ 3 × ℝ + ,

to describe an electron trapped in its own hole as an approximation to Hartree-Fock theory for a one component plasma. It also arises in multiple particle systems and Quantum Mechanics [2] . In a pioneering work, set ψ ( x , t ) = e i t u ( x ) , then

− Δ u + u = ( I 2 ∗ | u | 2 ) u ,     in     ℝ 3 , (1.3)

and Lieb [1] first obtained the existence and uniqueness of a ground state solution to (1.3) via variational methods. Lions [3] [4] considered the same problem and proved the existence and multiplicity of normalized solutions. The classification of positive solutions was first studied by Ma and Zhao [5] .

As for (1.2), Moroz and Van Schaftingen [6] studied the positivity, regularity, decay asymptotics and radial symmetry of ground state solutions for

N − 2 N + μ 1 < 1 p < N N + μ 1 . Meanwhile, they also proved that (1.2) has no nontrivial smooth H 1 solution for either 1 p ≤ N − 2 N + μ 1 or 1 p ≥ N N + μ 1 by using the Pohožaev identity. The number N + μ 1 N and N + μ 1 N − 2 (if N ≥ 3 ) are called the lower and upper critical exponents related to the Hardy-Littlewood-Sobolev inequality, respectively. Furthermore, if μ 1 = μ 2 , 0 ≤ α = β ≤ μ 1 2 and p = q , u = v in (1.1), Du et al. [7] also established the nonexistence result for

p ≥ N + μ 1 − 2 α N − 2 or p ≤ N + μ 1 − 2 α N when N ≥ 3 by the Pohožaev identity.

By using the method of moving planes in integral forms introduced by Chen et

al. [8] , Le [9] proved the following equation has no positive solution if p < N + μ 1 N − 2 ( p ∈ ℝ when N ≤ 2 ), and every positive solution u has the form u ( x ) = c ( λ λ 2 + | x − x 0 | 2 ) N − 2 2 for some c , λ > 0 and x 0 ∈ ℝ N ,

− Δ u = ( I μ 1 ∗ | u | p ) | u | p − 2 u ,     in     ℝ N .

As for more investigations about Choquard equations, we refer to [10] .

For the Hamiltonian system, if α = β = 0 in (1.1), Maia and Miyagaki [11] studied the following Choquard-type Hamiltonian system

{ − Δ u + u = ( I μ 1 ∗ | v | p ) | v | p − 2 v ,     in     ℝ N , − Δ v + v = ( I μ 2 ∗ | u | q ) | u | q − 2 u ,     in     ℝ N , u ( x ) , v ( x ) → 0,   when       | x | → ∞ . (1.4)

However, different from (1.2), the structure of (1.4) makes it quite difficult to obtain the Pohožaev identity for (1.4). In the spirit of the method in [12] , Maia and Miyagaki used similar arguments to overcome this difficulty and obtained the Pohožaev identity, they proved that: if N ≥ 3 , (1.4) has no nontrivial non-negative C 2 solution for p ≥ N + μ 1 N − 2 and q ≥ N + μ 2 N − 2 ; if N = 2 , (1.4) has no nontrivial non-negative C 2 solution for p ≤ 2 + μ 1 2 and q ≤ 2 + μ 2 2 . The key

idea of [12] is that, for the Hamiltonian system of 2 equations, consider a pair of non-negative solutions ( u , v ) , define U : = x ⋅ ∇ u ( x ) and V : = x ⋅ ∇ v ( x ) . Through a straightforward calculation to obtain v Δ U and u Δ V , then by using the differential knowledge and ( u , v ) is solution, we can get a differential form of Pohožaev identity, which will produce the Pohožaev identity. This new idea also helps Kou and An [13] to generalize the well-known results of Mitidieri [14] and discuss the nonexistence result of positive solutions for the Hamiltonian system in a non-star shaped domain. By the method of moving planes in integral forms, Le [15] also showed that the following system has no positive classical solution

{ − Δ u = ( I μ 1 ∗ | v | p ) | v | p − 2 v ,     in     ℝ N , − Δ v = ( I μ 2 ∗ | u | q ) | u | q − 2 u ,     in     ℝ N ,

when N ≥ 3 , 1 < p ≤ N + μ 1 N − 2 , 1 < q ≤ N + μ 2 N − 2 and p + q < 2 N + μ 1 + μ 2 N − 2 .

Other existence or nonexistence results of solutions for equations can be find, we refer readers to [16] [17] [18] [19] [20] and references therein.

Motivated by the aforementioned papers, in the present paper, we give a nonexistence result of nontrivial non-negative solutions for (1.1) with α , β ≥ 0 by the Pohožaev identity.

The main result of this paper is the following:

Theorem 1.1. Assume that N ≥ 3 , 0 < μ 1 , μ 2 < N , 0 ≤ α ≤ μ 1 2 , 0 ≤ β ≤ μ 2 2 , p , q > 1 , and N + μ 1 − 2 α p + N + μ 2 − 2 β q ≤ 2 ( N − 2 ) , if

( u , v ) ∈ ( C 2 ( ℝ N \ { 0 } ) ∩ H 1 ( ℝ N ) ∩ L 2 N p N + μ 1 − 2 α ( ℝ N ) )                       × ( C 2 ( ℝ N \ { 0 } ) ∩ H 1 ( ℝ N ) ∩ L 2 N q N + μ 2 − 2 β ( ℝ N ) ) is a pair of non-negative solutions of (1.1), then ( u , v ) = ( 0 , 0 ) . In particular, (1.1) does not have a pair of nontrivial non-negative solutions for p ≥ N + μ 1 − 2 α N − 2 and q ≥ N + μ 2 − 2 β N − 2 .

In Section 2, we will give the proof of Theorem 1.1. To facilitate reading, we use the notations:

• C 2 ( ℝ N \ { 0 } ) is the space of functions whose 2-th derivatives are continuous in ℝ N \ { 0 } .

• C 0 ∞ ( ℝ N ) is the space of functions infinitely differentiable with compact support in ℝ N .

• L p ( ℝ N ) , p ∈ [ 1, + ∞ ) is the usual Lebesgue space endowed with the norm ‖ u ‖ t = ( ∫ ℝ N | u | t d x ) 1 t .

• H 1 ( ℝ N ) = { u ∈ L 2 ( ℝ N ) : ∇ u ∈ L 2 ( ℝ N ) } is endowed with norm ‖ u ‖ = ( ∫ ℝ N | ∇ u | 2 + u 2 d x ) 1 2 .

To study (1.1), we need the following doubly weighted Hardy-Littlewood-Sobolev inequality proved in [21] .

Proposition 2.1. (Doubly Weighted Hardy-Littlewood-Sobolev Inequality) Let t , r > 1 and 0 < μ < N with 0 ≤ α + β ≤ μ , 1 t + α + β − μ N + 1 r = 1 , α < N t ′ , β < N r ′ , f ∈ L t ( ℝ N ) and h ∈ L r ( ℝ N ) , where 1 t + 1 t ′ = 1 and 1 r + 1 r ′ = 1 . Then there exists a constant C ( α , β , μ , N , t , r ) > 0 which is independent of f , h such that

∫ ℝ N ( I μ ∗ f ( x ) | x | α ) h ( x ) | x | β d x ≤ C ( α , β , μ , N , t , r ) ‖ f ‖ t ‖ h ‖ r .

From Proposition 2.1, we easily get the following remark.

Remark 2.1. Let 0 < μ < N , 0 ≤ α ≤ μ 2 , and p > 1 . Assume that u ∈ L 2 N p N + μ − 2 α ( ℝ N ) , then there exists C ( α , μ , N ) > 0 such that

∫ ℝ N ( I μ ∗ | u | p | x | α ) | u | p | x | α d x ≤ C ( α , μ , N ) ‖ u ‖ 2 N p N + μ − 2 α 2 p . (2.1)

Consider a cut-off function φ ∈ C 0 ∞ ( ℝ N , [ 0,1 ] ) such that φ ( x ) = 1 if | x | ≤ 1 , φ ( x ) = 0 if | x | ≥ 2 . For any fixed λ > 0 , set

u ˜ λ ( x ) = φ ( λ x ) x ⋅ ∇ u ( x )     and     v ˜ λ ( x ) = φ ( λ x ) x ⋅ ∇ v ( x ) .

Lemma 2.1. Let u , v ≥ 0 and ( u , v ) ∈ ( C 2 ( ℝ N \ { 0 } ) ∩ L 2 N p N + μ 1 − 2 α ( ℝ N ) ) × ( C 2 ( ℝ N \ { 0 } ) ∩ L 2 N q N + μ 2 − 2 β ( ℝ N ) ) , then

l i m λ → 0 ∫ ℝ N ( I μ 1 ∗ v p | x | α ) v p − 1 v ˜ λ | x | α d x = − N − μ 1 + 2 α 2 p ∫ ℝ N ( I μ 1 ∗ v p | x | α ) v p | x | α d x ,

and

l i m λ → 0 ∫ ℝ N ( I μ 2 ∗ u q | x | β ) u q − 1 u ˜ λ | x | β d x = − N − μ 2 + 2 β 2 q ∫ ℝ N ( I μ 2 ∗ u q | x | β ) u q | x | β d x .

Proof. A direct computation, one has

∫ ℝ N ( I μ 1 ∗ v p | x | α ) v p − 1 v ˜ λ | x | α d x = ∫ ℝ N ∫ ℝ N | x − y | μ 1 − N | y | − α v p ( y ) | x | − α v p − 1 ( x ) φ ( λ x ) [ x ⋅ ∇ v ( x ) ] d x d y = ∫ ℝ N ∫ ℝ N | x − y | μ 1 − N | y | − α v p ( y ) | x | − α φ ( λ x ) [ x ⋅ ∇ ( v p ( x ) p ) ] d x d y = 1 2 ∫ ℝ N ∫ ℝ N | x − y | μ 1 − N | x | − α | y | − α v p ( y ) φ ( λ x ) [ x ⋅ ∇ ( v p ( x ) p ) ]       + | x − y | μ 1 − N | x | − α | y | − α v p ( x ) φ ( λ y ) [ y ⋅ ∇ ( v p ( y ) p ) ] d x d y

= − 1 2 ∫ ℝ N ∫ ℝ N [ ( μ 1 − N ) | x − y | μ 1 − N | x | − α | y | − α v p ( y ) φ ( λ x ) v p ( x ) p x ( x − y ) | x − y | 2       + ( μ 1 − N ) | x − y | μ 1 − N | x | − α | y | − α v p ( x ) φ ( λ y ) v p ( y ) p − y ( x − y ) | x − y | 2       − α | x − y | μ 1 − N | x | − α | y | − α v p ( y ) φ ( λ x ) v p ( x ) p       − α | x − y | μ 1 − N | x | − α | y | − α v p ( x ) φ ( λ y ) v p ( y ) p

  + λ | x − y | μ 1 − N | x | − α | y | − α v p ( y ) [ x ⋅ ∇ φ ( λ x ) ] v p ( x ) p   + λ | x − y | μ 1 − N | x | − α | y | − α v p ( x ) [ y ⋅ ∇ φ ( λ y ) ] v p ( y ) p   + N | x − y | μ 1 − N | x | − α | y | − α v p ( y ) φ ( λ x ) v p ( x ) p   + N | x − y | μ 1 − N | x | − α | y | − α v p ( x ) φ ( λ y ) v p ( y ) p ] d x d y

= N − μ 1 2 p ∫ ℝ N ∫ ℝ N 1 | x − y | N − μ 1 v p ( x ) | x | α v p ( y ) | y | α ( x − y ) ⋅ ( x φ ( λ x ) − y φ ( λ y ) ) | x − y | 2 d x d y       + α p ∫ ℝ N ∫ ℝ N 1 | x − y | N − μ 1 v p ( x ) | x | α v p ( y ) | y | α φ ( λ x ) d x d y       − N p ∫ ℝ N ∫ ℝ N 1 | x − y | N − μ 1 v p ( x ) | x | α v p ( y ) | y | α φ ( λ x ) d x d y       − λ p ∫ ℝ N ∫ ℝ N 1 | x − y | N − μ 1 v p ( x ) | x | α v p ( y ) | y | α [ x ⋅ ∇ φ ( λ x ) ] d x d y .

Using the Lebesgue dominated convergence theorem, we get the first equality. Similarly, we obtain the second equality.

Lemma 2.2. Under the assumptions of Theorem 1.1, the following identity holds

4 ∫ ℝ N   u v d x = [ N + μ 1 − 2 α p − ( N − 2 ) ] ∫ ℝ N ( I μ 1 ∗ v p | x | α ) v p | x | α d x     + [ N + μ 2 − 2 β q − ( N − 2 ) ] ∫ ℝ N ( I μ 2 ∗ u q | x | β ) u q | x | β d x .

Proof. A direct computation, we have

v Δ u ˜ λ = λ 2 ( x ⋅ ∇ u ) v Δ φ ( λ x ) + 2 λ v ∇ φ ( λ x ) ⋅ ∇ ( x ⋅ ∇ u ) + φ ( λ x ) v Δ ( x ⋅ ∇ u ) ,

φ ( λ x ) v Δ ( x ⋅ ∇ u ) = φ ( λ x ) v [ 2 Δ u + x ⋅ ∇ ( Δ u ) ] ,

and

∂ ∂ x i [ ( I μ 1 ∗ v p | x | α ) v p | x | α x i ] = ( μ 1 − N ) ∫ ℝ N I μ 1 ( x − y ) v p ( y ) v p ( x ) x i ( x i − y i ) | x − y | 2 | x | α | y | α d y       + p ( I μ 1 ∗ v p | x | α ) v p − 1 | x | α v x i x i − α ( I μ 1 ∗ v p | x | α ) v p | x | α x i 2 | x | 2 + ( I μ 1 ∗ v p | x | α ) v p | x | α . (2.2)

Since ( u , v ) solves (1.1), we have − Δ u + u = ( I μ 1 ∗ v p | x | α ) v p − 1 | x | α , combing this with (2.2), we obtain

∫ ℝ N   v Δ u ˜ λ d x = λ 2 ∫ ℝ N   ( x ⋅ ∇ u ) v Δ φ ( λ x ) d x + 2 λ ∫ ℝ N   v ∇ φ ( λ x ) ⋅ ∇ ( x ⋅ ∇ u ) d x       + 2 ∫ ℝ N   φ ( λ x ) u v d x − 2 ∫ ℝ N   φ ( λ x ) ( I μ 1 ∗ v p | x | α ) v p | x | α d x       + ∫ ℝ N   v u ˜ λ ( x ) d x − ∑ i = 1 N   ∫ ℝ N   φ ( λ x ) ∂ ∂ x i [ ( I μ 1 ∗ v p | x | α ) v p | x | α x i ] d x       + ∑ i = 1 N   ∫ ℝ N   φ ( λ x ) ( I μ 1 ∗ v p | x | α ) v p − 1 | x | α v x i x i d x + N ∫ ℝ N φ ( λ x ) ( I μ 1 ∗ v p | x | α ) v p | x | α d x . (2.3)

Analogously,

∫ ℝ N   u Δ v ˜ λ d x = λ 2 ∫ ℝ N   ( x ⋅ ∇ v ) u Δ φ ( λ x ) d x + 2 λ ∫ ℝ N   u ∇ φ ( λ x ) ⋅ ∇ ( x ⋅ ∇ v ) d x       + 2 ∫ ℝ N   φ ( λ x ) u v d x − 2 ∫ ℝ N   φ ( λ x ) ( I μ 2 ∗ u q | x | β ) u q | x | β d x

      + ∫ ℝ N   u v ˜ λ ( x ) d x − ∑ i = 1 N   ∫ ℝ N   φ ( λ x ) ∂ ∂ x i [ ( I μ 2 ∗ u q | x | β ) u q | x | β x i ] d x       + ∑ i = 1 N   ∫ ℝ N   φ ( λ x ) ( I μ 2 ∗ u q | x | β ) u q − 1 | x | β u x i x i d x + N ∫ ℝ N   φ ( λ x ) ( I μ 2 ∗ u q | x | β ) u q | x | β d x . (2.4)

We multiply the first equation in (1.1) by v ˜ λ , and integrate over ℝ N , subtract from (2.3) to obtain

2 ∫ ℝ N φ ( λ x ) ( I μ 1 ∗ v p | x | α ) v p | x | α d x − 2 ∫ ℝ N   φ ( λ x ) u v d x = − ∫ ℝ N   v Δ u ˜ λ d x + λ 2 ∫ ℝ N   ( x ⋅ ∇ u ) v Δ φ ( λ x ) d x       + 2 λ ∫ ℝ N   v ∇ φ ( λ x ) ⋅ ∇ ( x ⋅ ∇ u ) d x       − ∑ i = 1 N   ∫ ℝ N   φ ( λ x ) ∂ ∂ x i [ ( I μ 1 ∗ v p | x | α ) v p | x | α x i ] d x       + ∫ ℝ N   v u ˜ λ ( x ) d x + N ∫ ℝ N   φ ( λ x ) ( I μ 1 ∗ v p | x | α ) v p | x | α d x       + ∫ ℝ N   v ˜ λ Δ u d x − ∫ ℝ N   u v ˜ λ d x + 2 ∫ ℝ N ( I μ 1 ∗ v p | x | α ) v p − 1 | x | α v ˜ λ d x . (2.5)

Similarly,

2 ∫ ℝ N   φ ( λ x ) ( I μ 2 ∗ u q | x | β ) u q | x | β d x − 2 ∫ ℝ N   φ ( λ x ) u v d x = − ∫ ℝ N   u Δ v ˜ λ d x + λ 2 ∫ ℝ N   ( x ⋅ ∇ v ) u Δ φ ( λ x ) d x       + 2 λ ∫ ℝ N   u ∇ φ ( λ x ) ⋅ ∇ ( x ⋅ ∇ v ) d x       − ∑ i = 1 N   ∫ ℝ N   φ ( λ x ) ∂ ∂ x i [ ( I μ 2 ∗ u q | x | β ) u q | x | β x i ] d x       + ∫ ℝ N   u v ˜ λ ( x ) d x + N ∫ ℝ N φ ( λ x ) ( I μ 2 ∗ u q | x | β ) u q | x | β d x       + ∫ ℝ N   u ˜ λ Δ v d x − ∫ ℝ N   v u ˜ λ d x + 2 ∫ ℝ N ( I μ 2 ∗ u q | x | β ) u q − 1 | x | β u ˜ λ d x . (2.6)

Recalling that u ( x ) , v ( x ) → 0 when | x | → ∞ and φ ∈ C 0 ∞ ( ℝ N , [ 0,1 ] ) , by divergence theorem and Fubini theorem, we obtain

− ∫ ℝ N   v Δ u ˜ λ d x + ∫ ℝ N   v ˜ λ Δ u d x − ∫ ℝ N   u Δ v ˜ λ d x + ∫ ℝ N   u ˜ λ Δ v d x = 0

and

− ∑ i = 1 N   ∫ ℝ N   φ ( λ x ) ∂ ∂ x i [ ( I μ 1 ∗ v p | x | α ) v p | x | α x i ] d x − ∑ i = 1 N   ∫ ℝ N   φ ( λ x ) ∂ ∂ x i [ ( I μ 2 ∗ u q | x | β ) u q | x | β x i ] d x = λ ∫ ℝ N ( I μ 1 ∗ v p | x | α ) v p | x | α [ x ⋅ ∇ φ ( λ x ) ] d x + λ ∫ ℝ N ( I μ 2 ∗ u q | x | β ) u q | x | β [ x ⋅ ∇ φ ( λ x ) ] d x .

Consequently, adding (2.5) and (2.6), by Lemma 2.2, it follows that

2 ∫ ℝ N ( I μ 1 ∗ v p | x | α ) v p | x | α d x + 2 ∫ ℝ N ( I μ 2 ∗ u q | x | β ) u q | x | β d x − 4 ∫ ℝ N   u v d x = N ∫ ℝ N ( I μ 1 ∗ v p | x | α ) v p | x | α d x + − N − μ 1 + 2 α p ∫ ℝ N ( I μ 1 ∗ v p | x | α ) v p | x | α       + N ∫ ℝ N ( I μ 2 ∗ u q | x | β ) u q | x | β d x + − N − μ 2 + 2 β q ∫ ℝ N ( I μ 2 ∗ u q | x | β ) u q | x | β ,

which is equivalent to

4 ∫ ℝ N   u v d x = [ N + μ 1 − 2 α p − ( N − 2 ) ] ∫ ℝ N ( I μ 1 ∗ v p | x | α ) v p | x | α d x     + [ N + μ 2 − 2 β q − ( N − 2 ) ] ∫ ℝ N ( I μ 2 ∗ u q | x | β ) u q | x | β d x .

The proof of Lemma 2.2 is complete.

Proof of Theorem 1.1: If ( u , v ) ∈ ( C 2 ( ℝ N \ { 0 } ) ∩ H 1 ( ℝ N ) ∩ L 2 N p N + μ 1 − 2 α ( ℝ N ) )   × ( C 2 ( ℝ N \ { 0 } ) ∩ H 1 ( ℝ N ) ∩ L 2 N q N + μ 2 − 2 β ( ℝ N ) ) is a pair of non-negative solutions of (1.1), suppose that N + μ 1 − 2 α p + N + μ 2 − 2 β q ≤ 2 ( N − 2 ) , by Lemma 2.2, there holds

0 ≤ 4 ∫ ℝ N   u v d x = ( N + μ 1 − 2 α p + N + μ 2 − 2 β q − 2 N + 4 ) ∫ ℝ N ( I μ 1 ∗ v p | x | α ) v p | x | α d x ≤ 0,

which indicates v = 0 . Similarly, we can prove that u = 0 .

The author declares no conflicts of interest regarding the publication of this paper.

Wang, Z.X. (2023) A Nonexistence Result for Choquard-Type Hamiltonian System. Journal of Applied Mathematics and Physics, 11, 608-617. https://doi.org/10.4236/jamp.2023.113038